Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain

1 Introduction

Fractional calculus and fractional differential equations have been studied extensively during the last decades. Fractional derivatives provide a more excellent tool for the description of memory and hereditary properties of various materials and processes than integer derivatives. Engineers and scientists have developed new models that involve fractional differential equations. These models have been applied successfully in, e.g., physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. For a systematic development of the topic, we refer to the books [1]-[7]. As an important issue for the theory of fractional differential equations, the existence of solutions to kinds of boundary value problems has attracted many scholars attention, and lots of excellent results have been obtained by means of fixed point theorems, upper and lower solutions technique, and so forth. A variety of results on initial and boundary value problems of fractional differential equations and inclusions can easily be found in the literature on the topic. For some recent results, we can refer to [8]-[18] and references cited therein.

Boundary value problems on infinite intervals appear often in applied mathematics and physics. Due to the fact that an infinite interval is noncompact, the discussion about boundary value problem on the infinite intervals is more complicated. Results on the existence of solutions of boundary value problems on infinite intervals for differential, difference and integral equations may be found in the monographs [19,20]. For boundary value problems of fractional order on infinite intervals we refer to [21]-[25].

Many researchers have shown their interest in the study of systems of fractional differential equations. The motivation for those works stems from both the intensive development of the theory of fractional calculus itself and the applications. See for example [26]-[30] where systems for fractional differential equations were studied by using Banach contraction mapping principle and Schaefer’s fixed point theorem.

Recently in [31] we investigated the existence of positive solutions for fractional differential equations of Hadamard type, with integral boundary condition on infinite intervals

where D^α denotes the Hadamard fractional derivative of order α, η ∈ (1, ∞) and I^β_i is the Hadamard fractional integral of order β_i > 0, i = 1, 2,... , m and λ_i ≥ 0, i = 1, 2,... , m are given constants. For some recent results on positive solutions of fractional differential equations we refer to [32]-[37] and references cited therein.

In [38] the existence of positive solutions were studied for the following fractional system of differential equations subject to the nonlocal Riemann-Liouville fractional integral boundary conditions of the form

where D^Φ are Riemann-Liouville fractional derivatives of orders ϕ∈{p,q},f,g∈C([0,1]×R+2,R+),I^ϕ are Riemann-Liouville fractional integrals of order ϕ ∈ {γ_i, μ_j}, α_i, β_j > 0, i = 1,..., m, j = 1,..., n and the fixed constants 0 < η < ξ < 1.

In this paper we investigate the existence of positive solutions for the following fractional system of Hadamard differential equations subject to the fractional integral boundary conditions on an unbounded domain

where D^Φ are Hadamard fractional derivatives of orders Φ ∈ {p, q} with lower limit 1,f,g∈C([1,∞]×R+2,R+), I^Φ are Hadamard fractional integrals of order Φ ∈ {α_i, β_j} with lower limit 1, λ_i, σ_j > 0, i,..., m, j = 1,..., n.

Applying first the well-known Guo-Krasnoselskii’s fixed point theorem we obtain the existence of at least one positive solution. Next we prove the existence of at least three distinct nonnegative solutions by using Leggett-Williams fixed point theorem.

The rest of the paper is organized as follows: In section 2, we present some preliminaries and lemmas that will be used to prove our main results. We also obtain the corresponding Green’s function and some of its properties. The main results are formulated and proved in Section 3. Especially in Subsection 3.1 we prove the existence of at least one positive solution while, in Subsection 3.2, we prove the existence of at least three distinct nonnegative solutions.

Examples illustrating our results are presented in Section 4.

2 Background materials and preliminaries

In this section, we present some notations and definitions of Hadamard fractional calculus (see [4]) and present preliminary results needed in our proofs later.